Nonlocal anisotropic Riesz interactions with a physical confinement

TL;DR

This paper characterizes the minimizers of nonlocal anisotropic Riesz energies supported on ellipsoids across different regimes, revealing isotropy independence in some cases and anisotropy effects in others, with explicit potential formulas involving hypergeometric functions.

Contribution

It provides a complete characterization of minimizers for anisotropic Riesz energies in any dimension, including explicit potential formulas and regime-dependent anisotropy effects.

Findings

Minimizers are independent of anisotropy in the super-Coulombic and Coulombic regimes.

In the sub-Coulombic regime, anisotropy can influence the optimality of minimizers.

A general formula for the potential inside an ellipsoid involving hypergeometric functions is derived.

Abstract

In this work we fully characterize, in any space dimension, the minimizer of a class of nonlocal and anisotropic Riesz energies defined over probability measures supported on ellipsoids. In the super-Coulombic and Coulombic regime, we prove that the minimizer is independent of the anisotropy. In contrast, in the sub-Coulombic regime we show that this property fails: we exhibit an example of anisotropy for which the isotropic minimizer is not optimal. In order to prove our main result, we provide a formula for the potential inside an ellipsoid, valid in any space dimension and involving the hypergeometric function.